Blood pressure and cardiac output, or pressure and flow, respectively, in the aorta of a patient, define its hemodynamic state. This hemodynamic state may change on a short time scale of seconds and minutes requiring continuous or semi-continuous monitoring. Thus, instrumentation has been developed for over a century for measuring both blood pressure and flow on a continuous basis. Unfortunately, the measurement of cardiac output (i.e. flow) is almost impossible to perform in a safe and continuous way. In contrast, blood pressure can be measured in patients on a continuous basis by invasive means with a little risk, but not entirely without risk, and more recently also non-invasively with the per se known Finapres methodology. Hence, there is a need for a method to derive flow from pressure using a computation instead of a measurement.
When recording pulsatile blood pressure and flow simultaneously in experimental animals, it was observed that if flow went up so did blood pressure and when flow went down blood pressure went down. Both hemodynamic signals are thus coupled. From physics and engineering one knows that pressure and flow are related via an impedance: p=qZ, with p pressure, q flow, and Z impedance. The proper impedance to relate aortic flow to aortic pressure is referred to as aortic impedance. But it is hard if not impossible to determine the aortic impedance in an individual patient. In principle, the impedance can be derived from the pressure and flow asZ=p/q, but the flow (q) cannot be measured easily as a waveform. A possible approach is the use of suitable models.
Windkessel Model
The oldest model for the hemodynamic properties of the aorta is the so-called “Windkessel” model. The equation to compute a stroke volume from the contour of the pressure pulse according to the Windkessel model is as follows:Vs=C(p2−p1)(1+As/Ad)with Vs—a stroke volume, C—an aortic compliance defined as dV/dP, p2—a pressure at a dicrotic notch, p1 the diastolic pressure, As the integrated area under the systolic portion of the blood pressure curve, and Ad similarly the diastolic area. The dicrotic notch is a pulse that precedes a dicrotic wave, it being a pulse sequence comprising a double-beat sequence wherein a second beat is weaker than a first beat. It is a disadvantage of this model that the compliance C of the aorta must be known. In practice the compliance is an unknown variable. In the prior art, the compliance has been determined indirectly by calibrating this value. To this purpose, a cardiac output has been measured with a standard clinical technique such as Fick or indicator dilution, Qi. A stroke volume from an indicator dilution, Vsi, follows as Vsi=Qi/f, with f being the heart rate. The compliance C now follows as the ratio C=Vsi/Vs (C=1). Once calibrated, the method can be used to follow changes and trends in stroke volume, for monitoring purposes.
However, this method has been shown to be unreliable. Various studies have been performed in which the compliance has been calibrated, followed by administrating of a vasoactive drug to change blood pressure, heart rate and cardiac output. It appeared that the compliance C changed with the drugs given, in various directions. This yielded that the Windkessel method might not be useful in practice.
Uniform Tube or Waterhammer Model
Another hemodynamic model of the aorta is the uniform tube with characteristic impedance, or Zc model. It describes the relation between pulsatile p(t) pressure and pulsatile flow q(t) in a uniform tube, while ignoring any mean pressure component:q(t)=p(t)/Zc with q(t) the pulsatile flow waveform, p(t) pulsatile pressure, and Zc the aortic characteristic impedance. Integrating the pulsatile signals from diastolic pressure, pd, during systole (when blood is ejected from the heart) one obtains:Vs=1/Zc∫(p(t)−pd)dt 
In this equation, the impedance Zc is unknown and can only be determined with an individual patient by calibration with a clinical cardiac output method, as described above for the Windkessel method. When tested under the same circumstances as the Windkessel method, described above, Waterhammer method also appeared unreliable although to a lesser degree, since Zc can be written as:Zc=√(r/)(AC′))with r the density of blood, A the aortic cross-sectional area and C′ the compliance per unit length. When A increases, C′ decreases rendering their product and thus impedance Zc relatively constant.